Inter-chromosomal k-mer distances

Inversion Symmetry is a generalization of the second Chargaff rule, stating that the count of a string of k nucleotides on a single chromosomal strand equals the count of its inverse (reverse-complement) k-mer. It holds for many species, both eukaryotes and prokaryotes, for ranges of k which may vary from 7 to 10 as chromosomal lengths vary from 2Mbp to 200 Mbp. Building on this formalism we introduce the concept of k-mer distances between chromosomes. We formulate two k-mer distance measures, D₁ and D₂, which depend on k. D₁ takes into account all k-mers (for a single k) appearing on single strands of the two compared chromosomes, whereas D₂ takes into account both strands of each chromosome. Both measures reflect dissimilarities in global chromosomal structures.

After defining the various distance measures and summarizing their properties, we also define proximities that rely on the existence of synteny blocks between chromosomes of different bacterial strains. Comparing pairs of strains of bacteria, we find negative correlations between synteny proximities and k-mer distances, thus establishing the meaning of the latter as measures of evolutionary distances among bacterial strains. The synteny measures we use are appropriate for closely related bacterial strains, where considerable sections of chromosomes demonstrate high direct or reversed equality. These measures are not appropriate for comparing different bacteria or eukaryotes. K-mer structural distances can be defined for all species. Because of the arbitrariness of strand choices, we employ only the D₂ measure when comparing chromosomes of different species. The results for comparisons of various eukaryotes display interesting behavior which is partially consistent with conventional understanding of evolutionary genomics. In particular, we define ratios of minimal k-mer distances (KDR) between unmasked and masked chromosomes of two species, which correlate with both short and long evolutionary scales.

k-mer distances reflect dissimilarities among global chromosomal structures. They carry information which aggregates all mutations. As such they can complement traditional evolution studies , which mainly concentrate on coding regions.